#131886

order	512
nilpotency class	4
isoclinism	isoclinic class of #131796
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

596 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,6,7,4,3,5,8)(9,10,13,20,14,19,30,24,15,21,31,18,16,11,22,25)(12,23,26,39,32,17,33,35,28,37,29,38,27,36,40,34);(5,6)(7,8)(9,11)(10,16)(12,23)(13,20)(14,21)(15,19)(17,27)(18,31)(22,24)(25,30)(26,35)(28,37)(29,34)(32,36)(33,39)(38,40);(1,3,4,2)(5,7,6,8)(9,12)(10,17)(11,23)(13,29)(14,32)(15,28)(16,27)(18,35)(19,37)(20,34)(21,36)(22,33)(24,39)(25,38)(26,31)(30,40)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=hi,e^2=g,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dg,e^c=eg,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39082);